Solution 4.4:8a

From Förberedande kurs i matematik 1

If we use the formula for double angles, sin2x=2sinxcosx, and move all the terms over to the left-hand side, the equation becomes

2sinxcosx−2cosx=0.

Then, we see that we can take a factor cosx out of both terms,

cosx(2sinx−2)=0

and hence divide up the equation into two cases. The equation is satisfied either if cosx=0 or if 2sinx−2=0 .

cosx=0:

This equation has the general solution

x=2+n(n is an arbitrary integer).

2sinx−2=0 :

If we collect sinx on the left-hand side, we obtain the equation sinx=12 , which has the general solution

xx=4+2n=43+2n

where n is an arbitrary integer.

The complete solution of the equation is

xxx=4+2n=2+n=43+2n

where n is an arbitrary integer.